The interesting and non-obvious bit is that[[WikiPedia:Nyquist%E2%80%93Shannon_sampling_theorem|there's only one

+

[[Image:Dsat 008.png|360px|right]]

+

The interesting and non-obvious bit is that [[WikiPedia:Nyquist%E2%80%93Shannon_sampling_theorem|there's only one

bandlimited signal that passes exactly through each sample point]]; it's a unique solution. If you sample a bandlimited signal and then convert it back, the original input is also the only possible output.

bandlimited signal that passes exactly through each sample point]]; it's a unique solution. If you sample a bandlimited signal and then convert it back, the original input is also the only possible output.

−

+

A signal that differs even minutely from the original includes frequency content at or beyond Nyquist, breaks the bandlimiting requirement and isn't a valid solution.

−

[[Image:Dsat 008.png|360px|right]]

−

−

Before you say, "I can draw a different signal that passes through those points", if it differs even

−

minutely from the original, it includes frequency content at or beyond Nyquist, breaks the bandlimiting requirement and isn't a valid solution.

So how did everyone get confused and start thinking of digital signals as stairsteps? I can think of two good reasons.

So how did everyone get confused and start thinking of digital signals as stairsteps? I can think of two good reasons.

Revision as of 03:36, 26 February 2013

Wiki edition

Continuing in the "firehose" tradition of Episode 01, Xiph.Org's second video on digital media explores multiple facets of digital audio signals and how they really behave in the real world.

Demonstrations of sampling, quantization, bit-depth, and dither explore digital audio behavior on real audio equipment using both modern digital analysis and vintage analog bench equipment, just in case we can't trust those newfangled digital gizmos. You can download the source code for each demo and try it all for yourself!

“Of everything in the entire article, that was the number one thing
people wrote about. In fact, more than half the mail I got was questions and
comments about basic digital signal behavior. Since there's interest, let's
take a little time to play with some simple digital signals. ”

Veritas ex machina

If we pretend for a moment that we have no idea how digital signals really
behave, then it doesn't make sense for us to use digital test
equipment. Fortunately for this exercise, there's still plenty
of working analog lab equipment out there.

All of this equipment is vintage, but the specs are still quite good.
We start with the signal generator set to output a 1 kHz
sine wave at one VoltRMS.
We see the sine wave on the oscilloscope, can verify that it is indeed
1 kHz at 1 Volt RMS, which is 2.8 Volts
peak-to-peak,
and that matches the
measurement on the spectrum analyzer as well.

The analyzer also shows some low-level white noise
and just a bit of harmonic distortion,
with the highest peak about 70dB or so below
the fundamental.
This doesn't matter to the demos, but it's good to take notice of it now to avoid confusion later.

For digital conversion, we use a boring, consumer-grade, eMagic USB1
audio device. It's more than ten years old at this point, and it's
getting obsolete.

A recent converter can easily have an order of magnitude better specs.
Flatness,
linearity,
jitter,
noise behavior,
everything...
You may not
have noticed. Just because we can measure an improvement doesn't
mean we can hear it, and even these old consumer boxes were already at
the edge of ideal transparency.

The eMagic connects to my ThinkPad, which displays a digital
waveform and spectrum for comparison, then the ThinkPad
sends the digital signal right back out to the eMagic for
re-conversion to analog and observation on the output scopes.

Stairsteps

First demo: We begin by converting an analog signal to digital and
then right back to analog again with no other steps.

The signal generator is set to produce a 1kHz sine wave just like
before and we can see the analog sine wave on the input-side oscilloscope. The eMagic digitizes our signal to
16 bit PCM at 44.1kHz,
same as on a CD. The spectrum of the digitized signal on the Thinkpad matches what we saw earlier and what we see now on the analog spectrum analyzer, aside from its
high-impedance input being just a smidge noisier. For now, the waveform display shows our digitized sine wave as a
stairstep pattern, one step for each sample.

When we look at the output signal that's been converted
from digital back to analog, we see that it's exactly like the original sine wave. No stairsteps.

1 kHz is still a fairly low frequency, so perhaps the stairsteps are just
hard to see or they're being smoothed away. Next, set the signal generator to 15kHz, which is much closer to Nyquist.
Now the sine wave is represented by less than three samples per cycle, and the digital waveform appears rather poor! Yet the analog output is still a perfect sine wave, exactly like the original.
As we keep increasing frequency, all the way to 20kHz, the output waveform is still perfect. No jagged edges, no dropoff, no stairsteps.

So where'd the stairsteps go? It's a trick question; they were never there. Drawing a digital waveform as a stairstep was wrong to begin with.

A stairstep is a continuous-time function. It's jagged, and it's piecewise, but it has a defined value at every point in time.
A sampled signal is entirely different. It's discrete-time; it's only got a value right at each instantaneous sample point and it's
undefined, there is no value at all, everywhere between. A discrete-time signal is properly drawn as a lollipop graph.
The continuous, analog counterpart of a digital signal passes smoothly through each sample point, and that's just as true for high
frequencies as it is for low.

The interesting and non-obvious bit is that there's only one
bandlimited signal that passes exactly through each sample point; it's a unique solution. If you sample a bandlimited signal and then convert it back, the original input is also the only possible output.
A signal that differs even minutely from the original includes frequency content at or beyond Nyquist, breaks the bandlimiting requirement and isn't a valid solution.

So how did everyone get confused and start thinking of digital signals as stairsteps? I can think of two good reasons.

First: it's easy to convert a sampled signal to a true stairstep. Just
extend each sample value forward until the next sample period. This is
called a zero-order hold, and it's an important part of how some
digital-to-analog converters work, especially the simplest ones.
As a result, anyone who looks up digital-to-analog converter or
digital-to-analog conversion is probably going to see a diagram of a
stairstep waveform somewhere, but that's not a finished conversion,
and it's not the signal that comes out.

Second, and this is probably the more likely reason, engineers who
supposedly know better (yes, even I) draw stairsteps even though they're
technically wrong. It's a sort of one-dimensional version of
fat bits in an image editor.
Pixels aren't squares either, they're samples of a 2-dimensional
function space and so they're also, conceptually, infinitely small
points. Practically, it's a real pain in the ass to see or manipulate
infinitely small anything, so big squares it is.

Digital stairstep drawings are exactly the same thing. It's just a convenient drawing. The stairsteps aren't really there.

Bit-depth

When we convert a digital signal back to analog, the result is
also smooth regardless of the bit depth. 24 bits or 16 bits...
or 8 bits... it doesn't matter.

So does that mean that the digital bit depth makes no difference at
all? Of course not.

Channel 2 here is the same sine wave input, but we quantize with
dither down to 8 bits.

On the scope, we still see a nice
smooth sine wave on channel 2. Look very close, and you'll also see a
bit more noise. That's a clue.

If we look at the spectrum of the signal... aha! Our sine wave is
still there unaffected, but the noise level of the 8-bit signal on
the second channel is much higher!

That may have been hard to hear anything but the tone. Let's listen
to just the noise after we notch out the sine wave and then bring the
gain up a bit because the noise is quiet.

Those of you who have used analog recording equipment may have just
thought to yourselves, "My goodness! That sounds like tape hiss!"
Well, it doesn't just sound like tape hiss, it acts like it too, and
if we use a gaussian dither then it's
mathematically equivalent in every way. It is tape hiss.

Intuitively, that means that we can measure tape hiss and thus the noise floor
of magnetic audio tape
in bits instead of decibels, in order to put things in a
digital perspective. Compact cassettes (for those of you who are old enough to remember them) could reach as
deep as 9 bits in perfect conditions, though 5 to 6 bits was
more typical, especially if it was a recording made on a
tape deck. That's right... your mix tapes were only about 6 bits
deep... if you were lucky!

Dither

I keep saying that I'm quantizing with dither, so what is dither
exactly and, more importantly, what does it do?

The simple way to quantize a signal is to choose the digital
amplitude value closest to the original analog amplitude. Obvious,
right? Unfortunately, the exact noise you get from this simple
quantization scheme depends somewhat on the input signal,

so we may get noise that's inconsistent, or causes distortion, or is
undesirable in some other way.

Dither is specially-constructed noise that substitutes for the noise
produced by simple quantization. Dither doesn't drown out or mask
quantization noise, it actually replaces it with noise characteristics
of our choosing that aren't influenced by the input.

Let's watch what dither does. The signal generator has too much noise for this test so we'll produce a mathematically perfect sine wave with the ThinkPad and quantize it to 8 bits with dithering.

We see a nice sine wave on the waveform display and output scope and, once the analog spectrum analyzer catches up...
a clean frequency peak with a uniform noise floor on both spectral displays
just like before. Again, this is with dither.

Now I turn dithering off.

The quantization noise, that dither had spread out into a nice, flat noise
floor, piles up into harmonic distortion peaks. The noise floor is
lower, but the level of distortion becomes nonzero, and the distortion
peaks sit higher than the dithering noise did.

At 8 bits this effect is exaggerated. At 16 bits,
even without dither, harmonic distortion is going to be so low as to
be completely inaudible.

Still, we can use dither to eliminate it completely if we so choose.

Turning the dither off again for a moment, you'll notice that the
absolute level of distortion from undithered quantization stays
approximately constant regardless of the input amplitude.
But when the signal level drops below a half a bit, everything
quantizes to zero.

In a sense, everything quantizing to zero is just 100% distortion!
Dither eliminates this distortion too. We reenable dither
and ... there's our signal back at 1/4 bit, with our nice flat noise floor.

Our hearing is most sensitive in the midrange from 2kHz to 4kHz,
so that's where background noise is going to be the most obvious.
We can shape dithering noise away from sensitive frequencies to where
hearing is less sensitive, usually the highest frequencies.

16-bit dithering noise is normally much too quiet to hear at all, but
let's listen to our noise shaping example, again with the gain
brought way up...

Lastly, dithered quantization noise is higher power overall
than undithered quantization noise even when it sounds quieter, and
you can see that on a VU meter during passages of near-silence. But
dither isn't only an on or off choice. We can reduce the dither's
power to balance less noise against a bit of distortion to minimize
the overall effect.

We'll also modulate the input signal like this to show how a varying input affects the quantization noise. At
full dithering power, the noise is uniform, constant, and featureless
just like we expect:

As we reduce the dither's power, the input increasingly
affects the amplitude and the character of the quantization noise.
Shaped dither behaves similarly, but noise shaping lends one more nice
advantage. To make a long story short, it can use a somewhat lower
dither power before the input has as much effect on the output.

Despite all the time I just spent on dither, we're talking about
differences that start 100 decibels and more below full scale. Maybe
if the CD had been
14 bits as originally designed,
dither might be
more important. Maybe. At 16 bits, really, it's mostly a wash. You
can think of dither as an insurance policy that gives several extra
decibels of dynamic range just in case. The simple fact is, though, no
one ever ruined a great recording by not dithering the final master.

Bandlimitation and timing

We've been using sine waves. They're the obvious choice when what we
want to see is a system's behavior at a given isolated frequency. Now let's look at something a bit more complex. What should we expect to happen when I change the input to a square wave... The input scope confirms our 1kHz square wave. The output scope shows… Exactly what it should.

What is a square wave really?

Well, we can say it's a waveform that's some positive value for half a cycle and then transitions instantaneously to a negative value for the other half.

squarewave(t)={1,|t|<T10,T1<|t|≤12T

But that doesn't really tell us anything useful about how that input becomes this output.

In modern web browsers you can program audio synthesizers directly in javascript. Use the two square wave formulas to get a square wave out of this page. (Note: The scope is not very accurate/useful.)

The rippling you see around sharp edges in a bandlimited signal is called the Gibbs effect. It happens whenever you slice off part of the frequency domain in the middle of nonzero energy.

The usual rule of thumb you'll hear is "the sharper the cutoff, the
stronger the rippling", which is approximately true, but we have to be
careful how we think about it.

For example... what would you expect our quite sharp anti-aliasing filter
to do if I run our signal through it a second time?

Aside from adding a few fractional cycles of delay, the answer is...
nothing at all. The signal is already bandlimited. Bandlimiting it
again doesn't do anything. A second pass can't remove frequencies
that we already removed.

And that's important. People tend to think of the ripples as
a kind of artifact that's added by anti-aliasing and anti-imaging
filters, implying that the ripples get worse each time the signal
passes through. We can see that in this case that didn't happen. So
was it really the filter that added the ripples the first time
through? No, not really. It's a subtle distinction, but Gibbs effect
ripples aren't added by filters, they're just part of what a
bandlimited signal is.

Even if we synthetically construct what looks like a perfect digital
square wave,

it's still limited to the channel bandwidth. Remember,
the stairstep representation is misleading.

What we really have here are instantaneous sample points,

and only one bandlimited signal fits those points. All we did when we
drew our apparently perfect square wave was line up the sample points
just right so it appeared that there were no ripples if we played
connect-the-dots.

But the original bandlimited signal, complete with ripples, was
still there.

And that leads us to one more important point. You've probably heard
that the timing precision of a digital signal is limited by its sample
rate; put another way,

that digital signals can't represent anything that falls between the
samples.. implying that impulses or
fast attacks have to align exactly
with a sample, or the timing gets mangled... or they just disappear.

At this point, we can easily see why that's wrong.

Again, our input signals are bandlimited. And digital signals are
samples, not stairsteps, not 'connect-the-dots'. We most certainly
can, for example, put the rising edge of our bandlimited square wave
anywhere we want between samples.

It's represented perfectly and it's reconstructed perfectly.

Epilogue

Just like in the previous episode, we've covered a broad range of
topics, and yet barely scratched the surface of each one. If anything, my
sins of omission are greater this time around... but this is a good
stopping point.

Or maybe, a good starting point. Dig deeper. Experiment. I chose my
demos very carefully to be simple and give clear results. You can
reproduce every one of them on your own if you like. But let's face
it, sometimes we learn the most about a spiffy toy by breaking it open
and studying all the pieces that fall out. And that's OK, we're
engineers. Play with the demo parameters, hack up the code, set up
alternate experiments. The source code for everything, including the
little pushbutton demo application, is up at xiph.org.

In the course of experimentation, you're likely to run into something
that you didn't expect and can't explain. Don't worry! My earlier
snark aside, Wikipedia is fantastic for exactly this kind of casual
research. And, if you're really serious about understanding signals,
several universities have advanced materials online, such as the
6.003
and
RES.6-007
Signals and Systems modules at MIT OpenCourseWare. And of
course, there's always the community here at Xiph.Org.

Digging deeper or not, I am out of coffee, so, until next time, happy
hacking!

A Co-Production of Xiph.Org and Red Hat, Inc.
(C) 2012-2013, Some Rights Reserved

Use The Source Luke

As stated in the Epilogue, everything that appears in the video demos is driven by open source software, which means the source is both available for inspection and freely usable by the community. The Thinkpad that appears in the video was running Fedora 17 and Gnome Shell (Gnome 3). The demonstration software does not require Fedora specifically, but it does require Gnu/Linux to run in its current form. In all, the video involved just under 50,000 lines of new and custom-purpose code (including contributions to non-Xiph projects such as Cinelerra and Gromit).

The Spectrum and Waveform Viewer

The realtime software spectrum analyzer application that appears in the video was a preexisting application that was dusted off and updated for use in the video. The waveform viewer (effectively a simple software oscilloscope) was written from scratch making use of some of the internals from the spectrum analyzer application. Both are available from Xiph.Org svn:

Spectrum and Waveform both expect an input stream on the command line, either as raw data or as a WAV file.

GTK-Bounce

The touch-controlled application used in the video is named 'gtk-bounce' and was custom-written for the sole purpose of the in-video demonstrations. It is so named because, for the most part, all it does is read the input from an audio device, and then immediately write the same data back out for playback. It also forwards a copy of this data to up to two external monitoring applications, and in several demos, applies simple filters or generates simple waveforms. It includes several demos not included in the video.

Starting Gtk-bounce

The application is somewhat hardwired for specific demo parameters, but most of the hardwired settings can be found at the top of each source file. As found in SVN, the application expects an ALSA hardware audio device at hw:1, and if none if found, it will wait for one to appear. Once a sound device is successfully initialized, it expects to find and open two pipes named pipe0 and pipe1 for output in the current directory. In the video, the waveform and spectrum applications are started to take input from pipe0 and pipe1 respectively. The output sent to the two pipes is identical, and in most demos matches the output data sent to the hardware device for conversion to analog. The only exception is the tenth demo panel (which does not appear in the video) where gtk-bounce can be set to monitor the hardware inputs instead while the outputs are used to produce test waveforms.

Assuming gtk-bounce, spectrum and waveform have been checked out and built, the configuration seen in the video can be started using the following commands:

make the pipe fifos for the applications to communicate (only needs to be done once)

mkfifo pipe0 pipe1

start all three applications

waveform pipe0 & spectrum pipe1 & gtk-bounce &

Using Gtk-bounce

Gtk-bounce consists of eleven pushbutton panels (numbered zero through ten) that can be selected by scrolling up and dwon with the arrow buttons on the right side. Each panel is intended for a specific demo or part of a demo.

Panel 0: This panel presents buttons that allow the sound card to be configured in several sampling rates and bit depths. Samples read from the audio inputs are sent to the output pipes and audio outputs for playback without modification.

Panel 1: Both channels are forwarded to the outputs, however the user may select the bit depth of each channel independently. When the sound card is running in 16 bit mode and 16-bit depth is selected, the data is untouched. Requantization to a lower bit depth is performed with a flat triangle dither.

Panel 2: Both channels are re-quantized to the selected bit depth. Requantization to a lower bit depth is performed with a flat triangle dither.

Panel 3: 'generate sine wave' discards the audio inputs and instead internally generates a sine wave at 32 bit precision, which is then quantized to the selected bit depth, optionally with dither. The resulting signal is then forwarded to the output.

Panel 4: gtk-bounce generates a 16-bit sine wave of the selected amplitude, optionally with dither, and forwards the resulting signal to the outputs. The audio input from the audio device is discarded. Note that the slider sets the peak amplitude, not the peak-to-peak amplitude.

Panel 5: generates a 16-bit sine wave, optionally quantized using dither. The user may additionally select a flat or a shaped dither. The 'notch and gain' button applies a notch filter to the resulting signal, and boosts the gain of the remaining noise so that it's easily audible. The audio input from the audio device is discarded.

Panel 6: allows the user to play with the power of the dithering noise applied before quantizing the sine wave. Shaped or flat dither are available. The sine wave may also be modulated with a varying amplitude to highlight correlations between the input and the resulting quantization noise. The 'notch and gain' button applies a notch filter to the resulting signal, and boosts the gain of the remaining noise so that it's easily audible. The audio input from the audio device is discarded.

Panel 7: applies a sharper antialiasing (lowpass) filter than is likely to be built into the sound-card hardware (as there's generally no reason to use a filter quite this sharp in practice). The very sharp filter allows us to bandpass the demonstration square wave without any harmonics landing in the transition band. The input is read from the audio device, passed through this sharper filter, and then forwarded to the outputs.

Panel 8: when selected, generate a synthetic 'square wave' (this is not quite equivalent to a bandlimited analog square wave; the harmonic amplitudes are a bit different) that when aligned with the sampling phase just right gives the appearance of having infinite rise and fall time. The slider allows us to shift the waveform sample alignment back and forth by +/- one sample to reveal that the underlying signal is still band-limited.

Panel 9: as in panel 8, generate a 'perfect' synthetic 'square wave'. However, the slider now allows us to shift the sample alignment of the second channel with respect to the first, instead of shifting both channels. This allows us the trigger/lock the scope timing to the channel 1 waveform so we can see the fractional sample movement and alignment of the waveform on channel 2. The audio input from the audio device is discarded.

Panel 10: not used in the video; The audio device is configured to 24-bit input/output. The user may produce one of a range of test signals that are output to both the external applications and the audio device on the first channel. The input on the second channel is passed-through to the applications and audio device outputs unchanged. The first channel input is unused unless 'two input mode' is selected. When two input mode is selected, both input channels are read and the data sent to the external applications. Generated test signals are sent only to the audio hardware (on the first channel). This combination of test signals and input modes allows self-references frequency response, phase, noise, distortion and crosstalk testing of a given audio device.

Cairo Animations

The animations featured throughout the Episode 2 video were rapid-development spaghetti hack-jobs coded by hand in raw Cairo. Each module generated a series of PNG stills that were then stitched into an animation with Cinelerra or mplayer. In the interest of pointing and laughing at what really bad code looks like...